Problem: Evaluate the definite integral. $\int^{2}_{5}\left(\dfrac{12-x^3}{x^4}\right)\,dx = $ Choose 1 answer: Choose 1 answer: (Choice A) A $\ln\left(\dfrac52\right)-\dfrac{117}{250}$ (Choice B) B $-\ln\left(3\right)+\dfrac{62}{25}$ (Choice C) C $\ln\left(\dfrac13\right)-36$ (Choice D) D None of the above
Explanation: First, simplify and use the power and natural log rules: $\begin{aligned}\int^{2}_{5}\left(\dfrac{12-x^3}{x^4}\right)\,dx&=\int^{2}_{5}\left(\dfrac{12}{x^4}-\dfrac{x^3}{x^4}\right)\,dx \\&=~\int^{2}_{5}\left(12x^{-4}-x^{-1}\right)\,dx \\&=\left(-4x^{-3}-\ln(x)\right)\Bigg|^{2}_{{5}}\end{aligned}$ Second, plug in the limits of integration: $(-\ln({2})-4\cdot{2}^{-3})-(-\ln({5})-4\cdot{5}^{-3}) = \ln\left(\dfrac52\right)-\dfrac{117}{250}$. The answer: $\int^{2}_{5}\left(\dfrac{12-x^3}{x^4}\right)\,dx ~=~\ln\left(\dfrac52\right)-\dfrac{117}{250}$